View of A Model Theoretical Generalization of Steinitz’s Theorem

doi: 10.5007/1808-1711.2011v15n1p107

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Pontifícia Universidade Católica de São Paulo/Faculdade de São Bento

Abstract. Infinitary languages are used to prove that any strong isomorphism of substruc- tures of isomorphic structures can be extended to an isomorphism of the structures. If the structures are models of a theory that has quantifier elimination, any isomorphism of sub- structures is strong. This theorem is a partial generalization of Steinitz’s theorem for alge- braically closed fields and has as special case the analogous theorem for differentially closed fields. In this note, we announce results which will be proved elsewhere.

Keywords: Strong isomorphism; infinitary languages; isomorphism extension; quantifier elimination.

Dedicated to Newton C.A. da Costa on his 80^thbirthday, 2010, December.

Strong isomorphisms

Let L be a language and letE1 be an L-structure defined on a domain D₁. Given two infinite cardinals α,β,β ≤ α, and assuming that α is a regular cardinal, we shall denote byLαβ the infinitary language which has the same symbols as L and which allows sequences of conjunctions of less than α formulas and sequences of instantiations of less than β variables. We considerE1 also as aLαβ-structure and callL_αβ the extended language ofL.

Consider a second L-structure E2 defined on a domain D₂ and let Hi be L- substructures of Ei, defined on the domains F_i ⊂ D_i,i = 1, 2. The cardinals α,β being fixed, we shall denote by [ϕ]i ⊂ D_i^γ the relation defined by a formula ϕ of the languageL_αβ, of arity γ, in theL_αβ-structureEi,i=1, 2. For the definition of relation defined by a formula of an infinitary language see, for instance, Rodrigues et al. 2010.

Definition 1. An isomorphismh:F₁ →F₂ isL_αβ-strongif for any relationR⊂D₁^γ, which is defined by a formulaϕof arityγ < αof the languageLαβ we have

h^γ([ϕ]1∩F₁^γ) = [ϕ]2∩F₂^γ, Principia15(1): 107–110 (2011).

Published by NEL — Epistemology and Logic Research Group, Federal University of Santa Catarina (UFSC), Brazil.

108 Alexandre Martins Rodrigues & Edelcio de Souza where h^γ:F₁^γ→F₂^γ is the extension of the maphto the powerγof the domainF₁. IfhisL_ωω-strong, we say simply thathis astrong isomorphism.

Theorem 2. Let h:F₁→F₂be a strong isomorphism of the substructuresH1 andH2. Then, h isL_αβ-strong for any choice ofαandβ.

Proof. (Sketch). The proof is by induction on definition of formulas. If ϕ is an atomic formula, ϕ is finitary and the result holds by hypothesis. For negation and conjunction clauses the proves are straightforward. The only non trivial case is the instantiation which is proved by transfinite induction on the arity of formulas.

Ifh:F₁ →F₂ admits an extension ˜h:D₁→D₂ which is an isomorphism of the structuresE1andE2, then ˜h^γ([ϕ]1) = [ϕ]2, implying thathisL_αβ-strong. Theorem 3 below shows that ifαandβare large enough, the converse of this statement holds.

For the main ideas involved in the proof of theorem 3, see Sebastião e Silva 1985, fundamental theorem I, p. 109, and da Costa & Rodrigues 2007, theorem 7.1, p. 23.

The notion of strong isomorphisms seems due to Sebastião e Silva.

In the theorem below, d⁺ denotes the first cardinal greater than the cardinal d ofD₁.

Theorem 3. Assume that theL-structuresE1 andE2 are isomorphic and let h:F₁→ F₂ be anLd⁺d⁺-strong isomorphism ofH1 andH2. Then, there exists an isomorphism

˜h:D₁→D₂ofE1 andE2which extends h, that is,˜h–F₁=h.

Proof. (Sketch). Let ˜g : D₁ → D₂ an isomorphism of E1 andE2. Consider an enu- meration p:λ→F₁ ofF₁ and prolong ptoq:d →D₁ an enumeration of D₁, i.e., λ < d andq(i) = p(i) fori ∈λ. Let G be the automorphism group ofE1. Let ρ and ˜ρ be theG-orbit of pandq, respectively. Since that inLd⁺d⁺ every G-orbit of arity less thand⁺is definable (see, for instance, Rodrigues et al. 2010, theorem 3.7, p 127), there are formulasϕ and ˜ϕ definingρand ˜ρ, i.e.,ρ= [ϕ]1and ˜ρ= [ϕ]˜ 1. We may assume that the free variables ofϕ and ˜ϕ are Var(φ) ={x_i : i∈λ} and Var(φ) =˜ {x_i:i∈d}. Now, consider the formula

ψ((x_i)i∈λ):=∃η(ϕ˜∧ϕ).

whereηis the sequence of variables η:i∈d−λ7→x_i ∈Var. It is easy to see that p∈[ψ]1and thenρ⊂[ψ]1. SincehisLd⁺d⁺-strong, then

h^λ([ψ]1∩F₁^λ) = [ψ]2∩F₂^λ.

As p∈[ψ]1∩F₁^λ, we have h^λ(p)∈[ψ]2∩F₂^λ. Therefore, h^λ(p)∈[ψ]2 and then for every i ∈ d−λ, there exists c_i ∈ D₂ such that ((h(pi))i∈λ,(ci)i∈d−λ) ∈ [ϕ]˜ 2. Since [ϕ]˜ 2 is the orbit of ˜g◦q in E2, there exists g ∈ G such that ˜g◦g◦q = ((h(pi))i∈λ,(ci)i∈d−λ). We define ˜h= ˜g◦gand it is easy to see that ˜his an isomor- phism ofE1 andE2such that ˜h–F₁=h.

Principia15(1): 107–110 (2011).

A Model-Theoretical Generalization of Steinitz’s Theorem 109

Theorem 4 follows from theorems 2 and 3.

Theorem 4. If the L-structures E1 and E2 are isomorphic, any strong isomorphism h : F₁ → F₂ of two substructures H1 and H2 can be extended to an isomorphism

˜h:D₁→D₂ofE1 andE2.

Quantifier elimination and Extensions of Isomorphisms

Theorem 4 simplifies when applied to models of a theory that has quantifier elimi- nation.

Theorem 5. If E1 and E2 are isomorphic model of an L-theory that has quantifier elimination, any isomorphism of substructures H1 of E1 and H2 of E2 is a strong isomorphism.

Proof. (Sketch). By induction on the definition of formulas. Let h:F₁ → F₂ be an isomorphism of substructures H1 ofE1 and H2 ofE2. Ifϕ is atomic we have the result, for isomorphisms preserve primitive relations of the structures involved. Ifϕ is a negation or a conjunction and the arity ofϕisn, then the result follows from the fact thathⁿ preserves boolean operations. Ifϕ is an instantiation, as theL-theory has quantifier elimination, the result follows from the same reason.

The following theorem is an immediate consequence of theorems 4 and 5.

Theorem 6. If E1 and E2 are isomorphic model of an L-theory that has quantifier elimination, any isomorphism of substructuresH1ofE1andH2ofE2admits an exten- sion to an isomorphism ofE1andE2.

Theorem 6 is a model-theoretical generalization of well known theorems of the theories of algebraically closed fields and differentially closed fields of fixed charac- teristic.

References

Da Costa, N. C. A. & Rodrigues, A. A. M. 2007. Definability and invariance.Studia Logica86:

1–30.

Rodrigues, A. A. M.; de Miranda Filho, R. C.; de Souza, E. G. 2010. Definability in infinitary languages and invariance by automorphisms.Reports on Mathematical Logic45: 119–33.

Sebastião e Silva, J. 1985. On automorphisms of arbitrary mathematical systems. History and Philosophy of Logic6: 91–116.

Principia15(1): 107–110 (2011).

110 Alexandre Martins Rodrigues & Edelcio de Souza

ALEXANDREA. MARTINSRODRIGUES

Departamento de Matemática Universidade de São Paulo, USP São Paulo, SP

BRAZIL

[email protected] EDELCIOG.DESOUZA

Pontifícia Universidade Católica de São Paulo Programa de Estudos Pós-Graduados em Filosofia Departamento de Filosofia Faculdade de São Bento São Paulo, SP

BRAZIL

[email protected]

Resumo. Linguagens infinitárias são utilizadas para provar que qualquer isomorfismo forte de subestruturas de estruturas isomorfas pode ser estendido para um isomorfismo das estru- turas. Se as estruturas são modelos de teorias que admitem eliminação de quantificadores, qualquer isomorfismo de subestruturas é forte. Este teorema é uma generalização parcial do teorema de Steinitz para corpos algebricamente fechados e tem como caso especial o teo- rema análogo para os corpos diferencialmente fechados. Nesta nota, anunciamos resultados que serão demonstrados em um trabalho posterior.

Palavras-chave:Isomorfismo forte; linguagens infinitárias; extensão de isomorfismo; elimi- nação de quantificadores.

Principia15(1): 107–110 (2011).